Question

Show that the canonical norm of \(\ell_{1}\) is nowhere Fréchet differentiable and is Gâteaux differentiable at \(x=\left(x_{i}\right)\) iff \(x_{i} \neq 0\) for every \(i\). If \(\Gamma\) is uncountable, show that the canonical norm of \(\ell_{1}(\Gamma)\) is not Gâteaux differentiable at any point. Hint: Let \(x \in S_{X} .\) Given \(\varepsilon>0\), find \(i\) such that \(\left|x_{i}\right|<\varepsilon / 2\) and consider \(h=\varepsilon e_{i} .\) Show that \(\|x \pm h\| \geq 1+\varepsilon / 2\) and use Lemma \(8.3\). Note that every vector in \(\ell_{1}(\Gamma)\) has a countable support, choose a standard unit vector outside this support, and use Lemma \(8.3 .\)

Short answer

The canonical norm of \(\br\ell_1\br)\ is nowhere Fréchet differentiable and Gâteaux differentiable iff \(x_i \eq 0 \forall i\); the \(\br\ell_1(\Gamma)\br)\ norm is nowhere Gâteaux differentiable if \(\br\Gamma\br)\ is uncountable.

Step-by-step solution

Understand the canonical norm in \(\ell_1\)

The canonical norm in \(\ell_1\) space is given by \[ ||x||_1 = \sum_{i} |x_i| \] for a vector \(x = (x_i) \).

Definition of Fréchet Differentiability

A function is Fréchet differentiable at a point if it can be closely approximated by a linear map. For the norm in \(\ell_1\), this involves checking if there is a linear map that approximates the norm function at every point.

Show non-differentiability in \(\ell_1\)

To show that the \(\ell_1\) norm is nowhere Fréchet differentiable, consider any vector \(x \in \ell_1\). Suppose there exists a point where it is differentiable; we can then find a linear map that approximates the norm function at that point. However, the absolute value function \(|x_i|\) is not differentiable at 0 (which is necessarily involved due to the \(\ell_1\) structure). Thus, no such linear map consistently exists.

Definition of Gâteaux Differentiability

A function is Gâteaux differentiable at a point \(x\) if the directional derivative exists in every direction. For the norm in \(\ell_1\), this means checking if the directional derivatives \(\frac{d}{dt} \|x + th\|_1 |_{t=0}\) exist for all vectors \(h\).

Gâteaux Differentiability at Specific Points

Given that Gâteaux differentiability at a point \(x = (x_i)\) requires each \(x_i eq 0\), the absolute values behave smoothly away from zero. Hence, the directional derivative exists and the norm is Gâteaux differentiable iff \(x_i eq 0 \forall i\).

Uncountable Set \(\brGamma\) and Non-differentiability

If \(\Gamma\) is uncountable, consider a vector \(x \in S_X\) in \(\ell_1(\Gamma)\). By the problem hint, choose \(i \in \brGamma\) such that \(|x_i| < \frac{\brvarepsilon}{2}\) and consider \(h = \brvarepsilon e_i\). The norm modification suggests \(\br|x \pm h\br| \geq 1 + \frac{\brvarepsilon}{2}\), violating the condition for directional derivatives to exist uniformly.

Application of Lemma 8.3

According to Lemma 8.3, since every vector in \(\ell_1(\brGamma)\) has a countable support and the arguments extend from \(\ell_1\) detail, we confirm the norm doesn't meet Gâteaux differentiability criteria at any point.

Key concepts

Fréchet Differentiability

Let's start by understanding what Fréchet differentiability means. A function is Fréchet differentiable at a point if it can be closely approximated by a linear map at that point. This means that for a given function, there should exist a linear map that approximates how the function behaves very closely.

In mathematical terms, if we have a function \( f \), it is Fréchet differentiable at a point \( x \) in a vector space if there exists a linear map \( A \) such that: \( \frac{||f(x+h) - f(x) - A(h)||}{||h||} \to 0 \text{ as } h \to 0 \).

For the canonical norm in \( \text{ℓ}_{1} \) space, denoted by \( ||x||_1 = \sum_{i} |x_i| \), showing Fréchet differentiability involves checking if there is such a linear map that can approximate the norm function at every point. Due to the nature of the absolute value function \( |x_i| \), particularly around 0, we find that the \( \text{ℓ}_{1} \) norm is nowhere Fréchet differentiable.

This is because the absolute value function is not differentiable at 0, making it impossible to find a consistent linear approximation that works across the entire space. Thus, the canonical norm in \( \text{ℓ}_{1} \) space is nowhere Fréchet differentiable.

Gâteaux Differentiability

Now, let's move on to Gâteaux differentiability. A function is Gâteaux differentiable at a point if the directional derivative exists in every direction. This means that for a function to be Gâteaux differentiable at a point \( x \), the derivative in the direction of any vector \( h \) should exist.

Formally, for a function \( f \) to be Gâteaux differentiable at \( x \), it should satisfy: \(\frac{d}{dt} f(x + th)|_{t=0} \) for all vectors \( h \).

When considering the canonical norm in \( \text{ℓ}_{1} \) space, i.e., \( ||x||_1 = \sum_{i} |x_i| \), we check if the directional derivatives exist. Given that Gâteaux differentiability at a point \( x = (x_i) \) requires each \( x_i \) not equal to zero, the absolute value behaves smoothly away from zero. Thus, the canonical norm in \( \text{ℓ}_{1} \) space is Gâteaux differentiable at \( x \) if and only if \( x_i eq 0 \) for every \( i \).

In essence, if none of the components of the vector \( x \) are zero, then the norm function behaves well enough in every direction to be considered Gâteaux differentiable.

Canonical Norm

The canonical norm in \( \text{ℓ}_{1} \) space is a fundamental concept in functional analysis. It's defined for a vector \( x = (x_i) \) in \( \text{ℓ}_{1} \) space by the sum of the absolute values of its components: \( ||x||_1 = \sum_{i} |x_i| \).

This norm measures the 'length' or 'size' of the vector in a specific way by summing up the magnitudes of all its components. It's called the canonical norm because it is a standard way to define the norm in this space.

Understanding the behavior of the canonical norm is crucial for studying differentiability properties. In particular, challenges in differentiability arise around zero due to the non-differentiable nature of the absolute value function at zero.

For example, when checking for Gâteaux differentiability of the canonical norm, we see that issues occur when any component \( x_i \) of the vector is zero. The smooth behavior away from zero ensures differentiability in those regions, but not at or around zero.

Vector Space

Lastly, let's touch on the concept of a vector space, a central building block in functional analysis. A vector space is a collection of vectors, which can be added together and multiplied (scaled) by numbers, called scalars.

In more formal terms, a vector space over a field \( F \) (like the field of real numbers \( \text{ℝ} \)) is a set \( V \) equipped with two operations: vector addition and scalar multiplication. These operations must satisfy certain axioms such as associativity, commutativity, and distribution.

To give a concrete example, the set of all sequences of real numbers that converge to zero, denoted by \( \text{ℓ}_{1} \), forms a vector space. Vectors in \( \text{ℓ}_{1} \) can be added together, and scaled by real numbers, while satisfying the necessary axioms.

In this context, we often study norms like the canonical norm to measure the size or length of vectors within this space, thus allowing us to analyze properties such as convergence, continuity, and differentiability.